Really? Too bad he shows, like, zero evidence of it over the course of the series.

He actually shows evidence of it in the very next episode that Screed will be watching. He is a very capable engineer.

Most of the enlisted crew are probably better engineers, and most of the officers who out rank Harry are not engineers.

Being a good engineer means "little".

Being an officer is about telling people what to do well and doing what you're told well. It's about being an exemplary part of the machine. The machine of course is the millions of tons of flesh working the Federation Fleet.